Properties

Label 966.19
Modulus $966$
Conductor $161$
Order $66$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(966, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,55,45]))
 
pari: [g,chi] = znchar(Mod(19,966))
 

Basic properties

Modulus: \(966\)
Conductor: \(161\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{161}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 966.be

\(\chi_{966}(19,\cdot)\) \(\chi_{966}(61,\cdot)\) \(\chi_{966}(103,\cdot)\) \(\chi_{966}(145,\cdot)\) \(\chi_{966}(157,\cdot)\) \(\chi_{966}(199,\cdot)\) \(\chi_{966}(241,\cdot)\) \(\chi_{966}(283,\cdot)\) \(\chi_{966}(313,\cdot)\) \(\chi_{966}(355,\cdot)\) \(\chi_{966}(451,\cdot)\) \(\chi_{966}(481,\cdot)\) \(\chi_{966}(493,\cdot)\) \(\chi_{966}(523,\cdot)\) \(\chi_{966}(619,\cdot)\) \(\chi_{966}(649,\cdot)\) \(\chi_{966}(661,\cdot)\) \(\chi_{966}(733,\cdot)\) \(\chi_{966}(787,\cdot)\) \(\chi_{966}(871,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((323,829,925)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{15}{22}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{61}{66}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{15}{22}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 966 }(19,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{966}(19,\cdot)) = \sum_{r\in \Z/966\Z} \chi_{966}(19,r) e\left(\frac{r}{483}\right) = 9.739569415+8.1326986671i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 966 }(19,·),\chi_{ 966 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{966}(19,\cdot),\chi_{966}(1,\cdot)) = \sum_{r\in \Z/966\Z} \chi_{966}(19,r) \chi_{966}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 966 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{966}(19,·)) = \sum_{r \in \Z/966\Z} \chi_{966}(19,r) e\left(\frac{1 r + 2 r^{-1}}{966}\right) = 8.0888287574+0.7723894961i \)